Until Kepler, astronomers had generally assumed that celestial objects moved in perfect circles. They argued that a perfect God would surely create objects that moved "perfectly". Kepler explained planetary movement with noncircular curves, known as ellipses. By using ellipses, Kepler discovered that he could make calculations that matched Brahe's observations. There are Kepler's Laws.

Kepler discovered the three laws of planetary motion while trying to achieve the Pythagorean[?] purpose of finding the harmony of the celestial spheres[?]. In his cosmovision, it was not a coincidence that the number of perfect polyhedra was one less than the number of known planets. Having embraced the Copernican system[?], he set out to prove that the distances from the planets to the sun where given by spheres inside perfect polyedra inside spheres.
He thereby identified the five platonic solids with the five intervals between the six known planets - Mercury, Venus, Earth, Mars, Jupiter, Saturn and the five classical elements.

In 1596 Kepler published The Cosmic Mystery . Here is a selection explaining the relation between the planets and the platonic solids:

... Before the universe was created, there were no numbers except the Trinity, which is God himself ... For, the line and the plane imply no numbers: here infinitude itself reigns. Let us consider, therefore, the solids. We must first eliminate the irregular solids, because we are only concerned with orderly creation. There remains six bodies, the sphere and the five regular polyhedra. To the sphere corresponds the heaven. On the other hand, the dynamic world is represented by the flat-faces solids. Of these there are five: when viewed as boundaries, however, these five determine six distinct things: hence the six planets that revolve about the sun. This is also the reason why there are but six planets ...

... I have further shown that the regular solids fall into two groups: three in one, and two in the other. To the larger group belongs, first of all, the Cube, then the Pyramid, and finally the Dodecahedron. To the second group belongs, first, the Octahedron, and second, the Icosahedron. That is why the most important portion of the universe, the Earth - where God's image is reflected in man - separates the two groups. For, as I have proved next, the solids of the first group must lie beyond the earth's orbit, and those of the second group within...Thus I was led to assign the Cube to Saturn, the Tetrahedron to Jupiter, the Dodecahedron to Mars, the Icosahedron to Venus, and Octahedron to Mercury ...

To emphasize his theory, Kepler envisaged an impressive model of the universe which shows a cube, inside a sphere, with a tetrahedron inscribed in it, another sphere inside it with a dodecahedron inscribed, a sphere with an icosahedron inscribed inside, and finally a sphere with an octahedron inscribed. Each of these celestial spheres had a planet embedded within them, and thus defined the planet's orbit.

On October 17, 1604, Kepler observed that an exceptionally bright star had suddenly appeared in the constellation Ophiuchus. (It had appeared on October 9 previous.) The appearance of the star, which Kepler described in his book De Stella nova in pede Serpentarii, provided further evidence that the cosmos was not changeless; this was to influence Galileo's argument. It has since been determined that the star was a supernova, the second in a generation, called Kepler's Star. No further supernovae have since been observed with certainty in the Milky Way, though others outside our galaxy have been seen.

In his 1619 book, Harmonice Mundi, as well as the treatise Misterium Cosmographicum, he also made an association between the Platonic solids with the classical conception of the elements: The tetrahedron was the form of fire, the octahedron was that of air, the cube was earth, the icosahedron was water, and the dodecahedron was the cosmos as a whole or ether. There is some evidence this association was of ancient origin, as Plato relates one Timaeus of Locri who thought of the Universe as being enveloped by a gigantic dodecahedron while the other four solids represent the "elements" of fire, air, earth, and water.

To his disappointment, Kepler's attempts to fix the orbits of the planets within a set of polyhedrons never worked out, but there were other rewards. Since he was the first to recognize the non-convex regular solids (such as the stellated dodecahedrons), they are named Kepler solids in his honor.

His most significant achievements came from the realisation that the orbits of the planets were ellipses, not circles. This realisation was a direct consequence of his failed attempt to fit the planetary orbits within polyhedra. Kepler's willingness to abandon his most cherished theory in the face of precise observational evidence indicates that he had a very modern attitude to scientific research, whereas previous generations of astronomers had been content to accept the authority of ideas from previous generations. Kepler also
made great steps in trying to describe the motion of the planets by appealing to
a force which resembled magnetism, and which emanated from the sun. Although he
did not discover gravity, he seems to have attempted to describe the first
empirical example of a universal law[?], to explain the behaviour of both earthly and heavenly bodies.

Kepler also made fundamental investigations into combinatorics, geometrical optimization, and natural phenomena such as snowflakes, always with an emphasis on form and design. He was also notable for defining antiprisms.